Czechoslovak Mathematical Journal, Vol. 68, No. 4, pp. 889-920, 2018
Linear FDEs in the frame of generalized ODEs: variation-of-constants formula
Rodolfo Collegari, Márcia Federson, Miguel Frasson
Received January 18, 2017. Published online August 24, 2018.
Abstract: We present a variation-of-constants formula for functional differential equations of the form
$\dot{y}={\mathcal L}(t)y_t+f(y_t,t), \quad y_{t_0}=\varphi,$
where ${\mathcal L}$ is a bounded linear operator and $\varphi$ is a regulated function. Unlike the result by G. Shanholt (1972), where the functions involved are continuous, the novelty here is that the application $t\mapsto f(y_t,t)$ is Kurzweil integrable with $t$ in an interval of $\mathbb R$, for each regulated function $y$. This means that $t\mapsto f(y_t,t)$ may admit not only many discontinuities, but it can also be highly oscillating and yet, we are able to obtain a variation-of-constants formula. Our main goal is achieved via theory of generalized ordinary differential equations introduced by J. Kurzweil (1957). As a matter of fact, we establish a variation-of-constants formula for general linear generalized ordinary differential equations in Banach spaces where the functions involved are Kurzweil integrable. We start by establishing a relation between the solutions of the Cauchy problem for a linear generalized ODE of type
$\frac{{\rm d}x}{{\rm d}\tau} = D[A(t)x],\quad x(t_0)=\widetilde{x}$
and the solutions of the perturbed Cauchy problem
$\frac{{\rm d}x}{{\rm d}\tau} = D[A(t)x+F(x,t)], \quad x(t_0)=\widetilde{x}.$
Then we prove that there exists a one-to-one correspondence between a certain class of linear generalized ODE and the Cauchy problem for a linear functional differential equations of the form
$\dot{y}={\mathcal L}(t)y_t, \quad y_{t_0}=\varphi,$
where $\mathcal L$ is a bounded linear operator and $\varphi$ is a regulated function. The main result comes as a consequence of such results. Finally, because of the extent of generalized ODEs, we are also able to describe the variation-of-constants formula for both impulsive FDEs and measure neutral FDEs.
Keywords: functional differential equation; variation-of-constants formula
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Affiliations: Rodolfo Collegari, Universidade Federal de Uberlândia, Av. João Naves de Ávila 2121, Santa Mônica, Uberlândia - MG, 38400-902, Brazil, e-mail: collegari@gmail.com; Márcia Federson, Miguel Frasson, Universidade de São Paulo, Av. Trab. São Carlense 400, Parque Arnold Schimidt, São Carlos - SP, 13566-590, Brazil, e-mail: federson@icmc.usp.br, frasson@icmc.usp.br
